UPDATES: The ring has been retrieved!!!! Thank you all for your advice, we really appreciate it
So we did almost all the things you all mentioned including:
1. Getting a endoscope camera and looking into all the vents
2. Taking out the air filter and checking inside
3. Cycling the air from defrost to front blow to low blow repeatedly ( I DO NOT RECOMMEND THIS i'll talk more about this in the NOTE section )
4. Taking apart the bottom portion of the car to see if it fell on the floor.
NOTE: This actually caused the ring to drop from the defroster deeper into the vents, so for all those reading the thread in the future - If you know that you dropped your ring into the defroster and your defroster was OFF at the time, DO NOT turn it on if you have a heavy ring. Chances are that it's still there and will stay there. It would have been a lot easier and maybe even possible for us to pop open the front speaker and use a hook or something to retrieve it, but I turned on the defroster and then heard a solid object falling behind the dashboard before i did that
Here's what happened next for those interested:
After we heard the ring fall, and watching a couple of dashboard disassembly videos, we knew that we probably wouldn't be able to do it ourselves. We aren't the handiest, so we decided to call up mazda. Save yourselves time and don't go here lol, especially if you're from NYC. They held my car for two days only to tell me after that they couldn't work on it for 2 more weeks. Spent like 100$ on ubers back and forth. They estimated 190 for a car inspection, and then 950 for dashboard removal.
So I get the car back after 2 days, make some sharp turns and go on bumpy roads to confirm that the mazda guys didn't take our ring, and we happen to hear it drop a little more in the center of the dashboard. We were like nice, at least it's still there. Called up a couple of local auto shops in my area, landed on H & B Auto Shop in Brooklyn with a guy named Benny who quoted me 700$ for dashboard removal and it would only take 2 days max.
I bring the car in during the afternoon wednesday, thursday afternoon I get a call from Benny saying that he found it! It was located in the evaporator, so taking apart the dashboard was a must since it was in that spot.
Anyways, lesson learned - we won't be taking off our rings anytime soon. Or putting anything small and valuable on the dashboard for that matter.
$800 well spent, ring retrieved, happy wife, happy life